We show:
The Boolean Prime Ideal theorem sans-serifBPI is equivalent to each one of the statements:
AC fin (: the axiom of choice restricted to families of finite sets) implies “every well ordered product of cofinite topologies is compact” and “every well ordered basic open cover of a product of cofinite topologies has a finite subcover”.
CAC fin (: the axiom of choice restricted to countable families of finite sets) iff “every countable product of cofinite topologies is compact”.
BPI(ω) (: every filter of ℘(ω) extends to an ultrafilter) is equivalent to the proposition “for every compact Loeb space boldY having a base of size ≤|R| and for every set X of size ≤|R| the product YX is compact”.